Welcome to Mrs. Wheaton's Blog! · Web viewFoundations of Algebra Day 6: One & Two Step Dimensional AnalysisNotes Foundations of Algebra Day 5: Metric Conversions & Defining Appropriate

Foundations of Algebra Module 3: Proportional Reasoning & Dimensional Analysis NotesModule 3: Proportional Reasoning &

Dimensional AnalysisAfter completion of this unit, you will be able to…

Learning Target #1: Proportional Reasoning with Ratios & Percents Represent ratios using models (Tables, Graphs, Double Number Lines) Use models to determine equivalent ratios Read and Interpret ratios from multiple representations Calculate unit rates and use them to interpret problems Explain the similarities and differences between percents, fractions, and decimals Convert between fractions, decimals, and percents Use mental math to calculate percents Determine the part, whole, or percent of a number Apply percents to real world problems (tax, tip, discounts)

Learning Target #2: Dimensional Analysis Convert units using dimensional analysis (Metric to Metric & customary to

customary) without conversion factor provided Convert units using dimensional analysis (between customary & Metric) with

conversion factor provided Define appropriate units for both metric and customary systems Apply dimensional analysis to rates

Timeline for Unit 2Monday Tuesday Wednesday Thursday Friday

10th Day 1 –

Equivalent Ratios

11th Day 2 –

Unit Rates & Their Graphs

12th Day 3 – Intro to

Percents

13th Day 4 – Percent

Problems

14th Day 5 –

Quiz over Days 1-4

Metric Conversions &

Appropriate Units

17th Day 6 -

1 & 2 Step Dimensional

Analysis

18th

Day 7 – Multi-Step

Dimensional Analysis

19th

Day 8 – Rate

ConversionsQuiz Over Days 5-8

20th Day 9 –

Review Day

21st

Day 10 –Module 3 Assessment

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Foundations of Algebra Day 1: Ratios & Proportions NotesDay

1: Ratios & Proportions

A ratio is a comparison of two nonnegative quantities that uses division. Ratios can compare part to part or part to whole relationships. Words that indicate ratio relationships are ______________________________________.

Consider the following scenario: On the co-ed soccer team, there are four times as many boys on it as it has girls. We would say the ratio is 4:1.

Part to Part Comparisons Part to Whole Comparisons

What other ratios would show four times as many boys as girls?

Practice: Create a ratio to describe the following:

a. There are 2 basketballs for every soccer ball.

b. There are 3 blueberry muffins in a 6 pack of muffins.

c. Each bagel costs $0.45.

d. For every 3 boys at soccer camp, there are 2 girls.

e. Billy wanted to write a ratio of the number of apples to the number of peppers in his refrigerator. He wrote 1:3. Did Billy write the ratio correctly?

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Standard(s): _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Foundations of Algebra Day 1: Ratios & Proportions Notes Rates vs Ratios

A rate is a ratio that compares two quantities that are measured in different units. If the rate is expressed as per 1 unit, it is considered a unit rate. When two ratios or rates are equivalent to each other, you can write them as a proportion. A proportion is an equation that states two ratios are equal.

Ratio

2 red rose: 5 white roses

Rate

90 miles: 2 hours

Unit Rate

45 miles: 1 hour

Proportion

Determine if the following can best be described as a ratio, rate, or unit rate:

a. 8 sugar cookies to 3 chocolate chip cookies b. 45 feet per second

c. 6 inches for every 3 years d. 6 boys for every 4 girls

Creating Equivalent Ratios by Scaling Up or Down

When we want to create equivalent ratios, we can use the same method as creating equivalent fractions. This is called scaling up or scaling down. Use the scaling up or scaling down method to determine the unknown quantity.

Creating Equivalent Ratios Using Tables

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Foundations of Algebra Day 1: Ratios & Proportions NotesWe can also use tables to determine equivalent ratios. Using the table below, show two calculations for the ratio of 150 lbs on Earth to 25 lbs on the moon.

Each table represents a series of equivalent ratios. Complete each table showing how you calculated each number. a.

b.

c.

Day 2: Proportions & Unit Rates and Their Graphs

4Standard(s): ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Foundations of Algebra Day 1: Ratios & Proportions NotesAs stated yesterday, a proportion states that two ratios are equal to each other. You spent yesterday creating equivalent rates using various models. A proportion allows you to create equivalent ratios using algebra. In order to solve proportions, you need to be able to solve a one-step equation.

Solve the following equations:

a. 3x = 9 b. 12x = 60 c. 2x = 10 d. 4x = 14

Creating Equivalent Ratios Using Proportions

When creating proportions, you can set up your proportions several ways. The key to creating them is to always match up corresponding parts or wholes. Take a look at the following scenario:

In a Valentine’s Day bouquet, 2 out of every 5 roses are pink. If there are 6 pink roses, how many total roses are in the bouquet?

Practice: Solve each problem by using a proportion.

a. Rita made 12 pairs of earrings in 2 hours. How many pairs of earrings could she make in 3 hours?

b. Perry earned $96 shoveling snow from 8 driveways. How much would Perry have earned if he had shoveled 10 driveways?

c. Marlene is planning a trip. She knows that her car gets 38 miles to the gallon on the highway. If her trip is going to be 274 miles and one gallon of gas is $2.30, how much should she expect to pay for gas?

Multi-Step with Proportions

a. For every 3 boys at soccer camp, there are 2 girls. If there are 20 children at soccer camp, how many are girls?

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Standard(s): ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Foundations of Algebra Day 1: Ratios & Proportions Notesb. It takes Ryan about 8 minutes to type a 500 word document. How long will it take him to type a 12 page essay with 275 words per page?

c. Josie took a long multiple choice test. The ratio of the number of problems she got incorrect to the number of problems she got correct was 2:9. If Josie missed 8 questions, how many did she get correct? How many questions were there total?

d. Sammy and David were selling water bottles to raise money for new football uniforms. Sammy sold 5 water bottles for every 3 water bottles David sold. Together, they sold 160 water bottles. How many did each boy sell?

e. The student faculty ratio at a small college is 17:3. The total number of students and faculty is 740. How many faculty and students are there at the college?

Unit Rates

A unit rate is a comparison of two quantities in which the denominator has a value of one unit. To calculate a unit rate, just divide the numerator by the denominator. Unit rates are helpful in real life for determining the best buy, most miles per gallon, the fastest car, cellphone, etc, and many other uses. Take a look at the following example:

A car dealership advertised the following rates on gal mileage for three new cars:

The Avalon can travel 480 miles on 10 gallons of gas.

The Compass can travel 400 miles on 8 gallons of gas.

The Patriot can travel 360 miles on 9 gallons of gas.

Which car gets the best gas mileage? Change each ratio to a unit rate to help make your decision.

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Foundations of Algebra Day 1: Ratios & Proportions Notes

Practice: Using unit rates, determine the best buy.

a.

b.

Unit rates are also helpful for calculating multiple numbers of an item (like when you are at the grocery store).

a. If a pound of bananas costs $0.53 a pound, how much are 4 pounds of bananas?

b. If a box of Cheerios costs $2.99, how much are 3 boxes of Cheerios?

c. If milk costs $2.59 a gallon, how much will 7 gallons cost?

Problem Solving with Unit Rates

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Foundations of Algebra Day 1: Ratios & Proportions Notesa. Anne is painting her house light blue. To make the color she wants, she must add 3 cans of white paint to every 2 cans of blue paint. How many cans of white paint will she need to mix with 6 cans of blue?

b. Ryan is making a fruit drink. The directions say to mix 5 cups of water with 2 scoops of powdered fruit mix. How many cups of water should he use with 9 scoops of fruit mix?

c. A publishing company is looking for new employees who can type at least 45 words per minute. Jessie can type 704 words in 16 minutes. Does she type fast enough to qualify for the job?

Using Unit Rates on a Graph

Claire & Kate entered a cup stacking contest so they have been practicing. Below is a graph of their progress.

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a. At what rate does each girl stack her cups during the practice session?

b. Kate notices she is not stacking her cups fast enough. What would Kate’s equation look like if she wanted to stack cups faster than Claire?

Foundations of Algebra Day 1: Ratios & Proportions NotesEmilio was to buy a new motorcycle. He wants to base his decision off the gas efficiency for each motorcycle. Which motorcycle is more gas efficient?

When viewing a unit rate on a graph, you are essentially looking at the ______________ of the line!!

Practice: Calculate the slope (unit rate) of each graph:a. b.

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c. d.

Day 3: Introduction to Percents

Robb’s Fruit Farm consists of 100 acres on which three different types of apples grow. On 25 acres, the farm grows Honeycrisp apples. McIntosh apples grow on 30% of the farm. The remainder of the farm grows Fuji apples. Shade in the grid below to represent the portion of the farm each type of apple occupies.

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Standard(s):______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Foundations of Algebra Day 1: Ratios & Proportions NotesType Color Fraction Decimal Percent

Honeycrisp

McIntosh

Fuji

Percents, fractions, and decimals can be used interchangeably. Percents are fractions that are out of 100. Percent is also another name for hundredths. The percent symbol “%” means out of 100. Percents are also considered ratios.

Converting Between Decimals, Percents, & Fractions

Percents to Decimals: a. 13% b. 6% c. 90% d. 125%

Decimals to Percents:a. 0.4 b. 0.32 c. 0.8427 d. 3.26

Fractions to Percents:

a. b. c. d.

Graphic Organizer for Converting Between Percents, Decimals, & Fractions

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Percents

35% means 35 out of 100.

35% as a fraction is .

35% as a decimal is 0.35.

35% as a ratio is 35 to 100 or 35:100.


Fraction Decimal Percent

Perc

ent

Write the percent as a fraction with a denominator of 100.

Move the decimal point two places to the left and remove the % sign.

Frac

tion

Divide the numerator by the denominator.

Use division to write the fraction as a decimal, and then convert to a percent (Move decimal two points to the right)

Dec

imal

Write the decimal as a fraction with a denominator of 10, 100, or 1000.

Move the decimal point two places to the right and add the % sign.

Calculating Percents of a Number with Common Percents

Discover: Use your calculator to determine the percent of each number:

a. 1% of 28 = ________ g. 10% of 28 = ________

b. 1% of 234 = ________ h. 10% of 234 = ________

c. 1% of 0.85 = ________ i. 10% of 0.85 = ________

d. 1% of 5.86 = ________ j. 10% of 5.86 = ________

e. 1% of 56.79 = ________ k. 10% of 56.79 = ________

f. 1% equals the decimal ______ l. 10% equals the decimal _______

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Calculating 1% of a Number

______________________________

______________________________


______________________________

______________________________


Practice: Calculate the following percents:

a. 1% of 90 = ________ b. 1% of 75 = ________ c. 1% of 200 =________

d. 10% of 90 = _________ e. 10% of 75 = _________ f. 10% of 200 = _________

There are certain percents, called benchmark percents that are used commonly in real life. They are 1%, 5%, 10%, 25%, 50%, and 100%. State each relationship below:

a. How is 50% related to 100%? d. How is 5% related to 10%?

b. How is 25% related to 100%? e. How is 1% related to 10%?

c. How is 10% related to 100% 50 f. How is 25% related to 50%?

Practice; Try calculating the following percents mentally.

Number 50% 10% 1% 5% 25%

300

50

400

16

13


______________________________

______________________________


______________________________

______________________________


Day 4: Percent Problems

Determining Parts, Wholes, & Percents

Percent problems involve three parts – the whole, the part, and the percent. As long as you know two out of the three quantities, you can determine the third. You can use the percent proportion to find the third quantity.

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Standard(s): _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________


Practice: Calculate the missing quantity using either double number lines or the percent proportion.

a. 25% of 48 is what number? b. 12 is 20% of what number?

c. 90 is 75% of what number? d. 42 is 30% of what number?

e. If Jackson paid $450 for a laptop that was 75% of the original price, what was the original price?

f. Eric once had 240 downloaded songs in his collection. He deleted some and now has 180. What percent of his original collection did he keep?

Percent Word Problems - Tax

The tax rate in your county is 7% of the subtotal, which is then added on to determine the final cost. Suppose you buy an item that costs $18.00. What will be your total cost?

Two Step Method One Step Method

Percent Word Problems - Tips

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Foundations of Algebra Day 1: Ratios & Proportions NotesYou and your friend go to your favorite restaurant, The Cheesecake Factory, this past weekend. It is customary that for good service you tip your waiter 15% of the bill and 20% for exceptional service. Your bill, before tips, was $45.00. You had good, not exceptional service. What will be your total bill?


Percent Word Problems - Discounts

Your favorite brand of shoes, Chacos, is having a big sale – 25% off all shoes. The shoes you really want are currently $105.00, but they will be included in the sale. How much are the shoes you want now?


Error Analysis: Explain what Katie did wrong and what the correct answer should be:

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Foundations of Algebra Day 1: Ratios & Proportions Notesa. Sandra got 4 problems wrong on a test of 36 questions. What percent of the questions did he get correct?

b. Games that usually sell for $36.40 were on sale for $27.30. What percent off are they?

c. Tahjama and Viva are shopping for new shoes. They notice a flyer that says 40% off the sales tag price of all shoes. Tahjama finds a pair of shoes she likes but there is no sticker that gives the final sale price of the shoes. She knows that the original price is $120 and the original sale price is 25% off. Tahjama thinks she can add the two sale percents together (25% + 40% = 65%) whereas Viva disagrees. Who is correct and what is the final price of the shoes?

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Foundations of Algebra Day 1: Ratios & Proportions Notesd. You need a graphing calculator for this class. The current price of the TI-84 Color calculators are regularly $120. Target, Staples, and Office Max are offering different sales on the TI-84 and you decide you want to save your parents as much money as possible. Which of the following offers will result in buying the calculator for the least amount of money?

Target has the price of the graphing calculator down 30%, but if you show your student id, you receive an additional 25% off the original price.

Staples has the price of the calculator marked down 25%, but if you come in between 1 pm and 3 pm, you get an additional 30% off the sale price.

Office Max has the price of the graphing calculator marked down 50%.

Which store has the best priced calculator and how much is the calculator at each store?

Day 5: Metric Conversions & Defining Appropriate Units

The Metric System of Measurement is based on multiples of 10. The three base units are meters, liters, and grams. The 6 prefixes are kilo (1000), hector (100), deka (10), base unit (1), deci (.1), centi (.01), and milli (.001). A helpful way to remember the order of the prefixes is King Henry Died Unusually Drinking Chocolate Milk.

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Standard(s): _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________


Examples: Convert from one prefix to anotherA. 2500 dL = __________ kL B. 38.2 dkg = __________ cg C. 5 dm = __________ m

D. 1000 mg = __________ g E. 14 km = __________ m F. 1 L = __________ mL

Examples: Compare measurements using <, >, or =. (Hint: They have to be written in the same units of measure before you can compare.)

A. 502 mm ________ .502 m B. 90,801 cg ________ 5 hg C. 160 dL ________ 1.6 L

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Foundations of Algebra Day 1: Ratios & Proportions Notes Defining Appropriate Units - Metric

Unit of Measure Abbreviation Estimate

Length

Millimeter mm 1 mm = thickness of a cd

Centimeter cm 1 cm = width of computer keyboard key

Meter m 1 m = length across a doorway

Kilometer km 1 km = length of 11 football fields

Mass

Milligram mg 1 mg = mass of a strand of hair

Gram g 1 g = mass of a dollar bill

Milligram kg 1 kg = mass of a textbook

Capacity

Milliliter mL 1 mL = sip from a drink

Liter L 1 L = amount of liquid in a bottle of water

Kiloliter kL 1 kL = amount of water in two bathtubs

Practice: Choose the appropriate metric unit of measure to use when measuring the following:

a. The length of your pencil:

b. The amount of water to fill a swimming poolc. Your height

d. The distance from New York to California

Defining Appropriate Units - Customary

Unit of Measure Abbreviation Estimate

Length

Inch in 1 in = length of small paper clip

Foot ft 1 ft = length of a man’s foot

Yard yd 1 yd = length across a doorway

Mile mi 1 mi = length of 4 football fields

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Foundations of Algebra Day 1: Ratios & Proportions NotesWeight

Ounce oz 1 oz = weight of one slice of cheese

Pound lb 1 lb = weight of one can of canned food

Ton t 1 t = weight of small car

Capacity

Fluid Ounce fl oz 1 fl oz = sip from a drink

Cup c 1 c = large scoop of ice cream

Pint pt 1 pt = school lunch milk container

Quart qt 1 qt = container of automobile oil

Gall gal 1 gal = large can of paint

Practice: Choose the appropriate metric unit of measure to use when measuring the following:

a. The height of a building

b. The weight of your biology textbook

c. The weight of a semi truck

d. The amount of chicken noodle soup in a soup can

e. The amount of water that fills a bathtub Defining Appropriate Units – Mixed Multiple Choice

1. Sandra collected data about the amount of rainfall a city received each week. Which value is MOST LIKELY part of Sandra's data?a) 3.5 feetb) 3.5 yardsc) 3.5 inchesd) 3.5 meters

2. What is a good unit to measure the area of a room in a house?a) Square feetb) Square milesc) Square inchesd) Square millimeters

3. If you were to measure the volume of an ice cube in your freezer, what would be a reasonable unit to use?a) Cubic feetb) Cubic miles

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Foundations of Algebra Day 1: Ratios & Proportions Notesc) Square feetd) Cubic inches

4. Which unit is the most appropriate for measuring the amount of water you drink in a day?a) Kilolitersb) Litersc) Megalitersd) Milliliters

Day 6: One & Two Step Dimensional Analysis

There are many different units of measure specific to the U.S. Customary System that you will need to remember. The list below summarizes some of the most important.

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Measurement Time Capacity

Weight

1 foot = ________ inches 1 minute = ________ seconds 1 cup = ________ fl. oz 1 ton = ________ lbs

1 yard = ________ feet 1 hour = ________ minutes 1 pint = ________ cups

1 lb = ________ oz

1 mile = ________ feet 1 day = ________ hours 1 quart = ________ pints

Standard(s): _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________


In order to convert between units, you must use a conversion factor. A conversion factor is a fraction in which the numerator and denominator represent the same quantity, but in different units of measure.

Examples: 3 feet = 1 yard: OR 100 centimeters = 1 meter: OR

Multiplying a quantity by a unit conversion factor changes only its units, not its value. It is the same thing as multiplying by 1.

The process of choosing an appropriate conversion factor is called dimensional analysis.

Understanding Dimensional Analysis

When setting up your conversion factors, don’t worry about the actual numbers until the very end. The key to set up your conversion factors so that they cancel out the units you don’t want until you end up with the units that you do want.

1. Convert from inches to miles

Possible Conversion Factors: yardsmiles

or milesyard

Inchesfeet

or feetinches

yardfeet

or feetyards

2. Convert from gallons to cups

Possible Conversion Factors: cupspints

or pintscups

quartspints

or pintsquarts

gallonsquarts

or quartsgallons

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Practicing Dimensional Analysis

Scenario 1 : How many feet are in 72 inches?Step 1: Write the given quantity with its unit of measure.Step 2: Set up a conversion factor.(Choose the conversion factor that cancels the units you have and replaced them with the units you want.

Step 3. Divide the units (only the desired unit should be left). Step 4: Solve the problem using multiplication and/or division.

Scenario 2: How many cups are in 140 pints?

Possible Conversion Factors:

Scenario 3: How many feet are in 4.5 miles?


Scenario 4: Convert 408 hours to days.


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Scenario 5: How many pounds are in 544 ounces?


Scenario 6: How many liters are in 4 quarts? (1.05 qt = 1 L)


Scenario 7: How many ounces are in 451 mL? (0.034 oz = 1 mL)


Video: Kendrick Farris clean and jerked 197 kg, 205 kg, and 211 kg at the 2013 Worlds Championships. How many pounds did he lift each time if 2.2 lbs = 1 kg?

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Day 7: Multi-Step Dimensional Analysis

How many seconds are in a day?

Most of us do not know how many seconds are in a day or hours in a year. However, most of us know that there are 60 seconds in a minute, 60 minutes in an hour, and 24 hours in a day. Some problems with converting units require multiple steps. When solving a problem that requires multiple conversions, it is helpful to create a flowchart of conversions you already know, set up your conversion factors, and solve your problem.

Flowchart: Days Hours Minutes Seconds

Conversion Factors: 60 sec = 1 min, 60 min = 1 hr 24 hours = 1 day

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Standard(s): _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________


Scenario 1: How many inches are in 3 miles?

Flowchart:

Possible Conversions:

Scenario 2: How many centimeters are in 900 feet? (2.54 cm = 1 in)

Flowchart:


Scenario 3: How many gallons are in 250 mL? (1 gal = 3.8 liters)

Flowchart:


Scenario 4: How many feet are in 5000 centimeters? (1 in = 2.54 cm)

Flowchart:


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Foundations of Algebra Day 1: Ratios & Proportions NotesReal World Applications

Scenario 5: One cereal bar has a mass of 37 grams. What is the mass of 6 cereal bars? Is that more or less than 1 kilogram?

Scenario 6: A rectangle has a length of 5 cm and 100 mm. What is the perimeter of the rectangle in millimeters?

Scenario 7: You’re throwing a pizza party for 15 people and figure that each person will eat 4 slices. Each pizza will cost $14.78 and will be cut up into 12 slices.

How many pizzas do you need for your party? How much will this cost?

Scenario 8: a. You find 13,406,190 pennies. How many dollars did you actually find?

b. If each penny weighs 4 grams, how much did all that loot weigh in lbs? (2.2 lbs = 1 kg)

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c. Assume a movie ticket costs $9. How many movie tickets could you buy with the pennies you found in part a?

Scenario 9: Mrs. Wheaton is approximately 280,320 hours old. How many years old is she?

Day 8: Rate Conversions

On Day 1, you learned what a rate is. Redefine what a rate is and then name a few examples.

Most of the rates we are going to discuss today include both an amount and a time frame such as miles per hour or words per minute. When we convert our rates, we are going to change the units in both the numerator and denominator.

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Rate

Examples:

Standard(s): _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________


a. Ms. Howard can run about 2 miles in 16 minutes. How fast is she running in miles per hour?

b. Convert 36 inches per second to miles per hour.

c. Convert 45 miles per hour to feet per minute.

d. Convert 32 feet per second to meters per minute. (Use 1 in = 2.54 cm)

e. A soccer ball deflates by 1 cm every 3 days. What is the rate of deflation in inches per week? (Use 1 in = 2.54 cm)

f. The top speed of a coyote is 43 miles per hour. Find the approximate speed in kilometers per minute. (Use 1 mile = 1,610 meters)

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g. The Washington family drinks 2 quarts of milk per day. How many gallons of milk do they drink in a week?

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Documents

Welcome to Mrs. Wheaton's Blog! · Web viewFoundations of Algebra Day 6: One & Two Step Dimensional AnalysisNotes Foundations of Algebra Day 5: Metric Conversions & Defining Appropriate